Huygens, Kirchhoff, Sommerfield and Rayleigh originated and contributed most to the currently accepted theory of diffraction, which forms the theoretical foundation for the present invention. Basically, the theory postulates that given a known wave front filling a planar window in an otherwise dark planar screen, the wave front at any point beyond the screen is calculable. Variations of this theory are used to compute the Fraunhofer far-field antenna pattern assuming a known field distribution at the antenna for electromagnetic wave fronts in the microwave range. An ordinary light camera, assuming a sufficiently coherent and pseudo monochromatic light wave, contains the Back Focal Plane (BFP) of the camera's objective lens as the illuminating window of the diffraction theory, and the imaging plane as the plane at which the image could be calculated. Of course, in the case of the camera, photographic film or electronic sensing devices are placed in the image plane, recording the intensity of the wave and no calculations need be made. However, it will be appreciated that at each point in a wave front there is more than just the intensity of the wave—there is the phase of the wave which may contain as much as eighty percent of the information about the object which is being imaged. To appreciate this fact more fully, it is only necessary to recall the potential of the conventional holograms to image objects in three dimensions. In particular, using phase information about a coherent wave front, holography creates three-dimensional images such that obscured objects may become visible if the observer steps to the side. So, the problem addressed using this invention can be stated as follows: given that a wave front is a complex function characterized by both amplitude (related in a straightforward manner to intensity) and phase at each point, how can the phase be captured using only intensity measurements.
To appreciate the complexity of the problem, consider the following observation: at a given instant of time, the phase of a wave length is about 6.28 radians. For red light, the length over which that phase is generated is about 0.6 microns. Considering that light travels at approximately 300,000,000 meters per second, that means that the frequency of such a wave passing a point in space is about 3.1*1015 radians/second. No device exists that has that kind of response time. For the ordinary light camera, the two planes of interest relating to diffraction theory are the BFP of the lens and the image plane. They have been shown to be conjugate planes in the sense that the wave front in the image plane is essentially the Fourier Transform of the illuminating wave in the BFP.
In a coherent monochromatic imaging system the problem of extracting phase information from a detection medium which records only intensity information remains a problem without a consistent solution. Several experimental methods have been proposed for determining the phase function across a wave front. One such method disclosed in Gabor, D. “A New Microscope Principle,” Nature 161, 777 (1948) involves the addition of a reference wave to the wave of interest in the recording plane. The resulting hologram records a series of intensity fringes, on a photographic plate, which contain enough information to reconstruct the complete wave function of interest. However, in most practical applications this method is cumbersome and impractical to employ.
Other methods, which do not employ reference waves, have been proposed for inferring the complete wave function from intensity recordings. See, e.g., Erickson, H. & Klug, A. “The Fourier Transform of an Electron Micrograph: Effects of Defocusing and Aberrations, and Implications for the use of Underfocus Contrast Enhancements”, Berichte der Bunsen Gesellschaft, Bd. 74, Nr. 11, 1129-1137 (1970). For the most part, these methods involve linear approximation and thus are only valid for small phase and/or amplitude deviations across the wave front of interest. In general, these methods also suffer from the drawback of requiring intensive computational resources.
A further method proposed that intensity recordings of wave fronts can be made conveniently in both the imaging and diffraction planes. Gerchberg, R. & Saxton, W. “Phase Determination from Image and Diffraction Plane Pictures in the Electron Microscope,” Optik, Vol. 34, No. 3, pp. 275-284 (1971). The method uses sets of quadratic equations that define the wave function across the wave in terms of its intensity in the image and diffraction planes. This method of analysis is not limited by the above-described deficiency of being valid for small phase or amplitude deviations, but again, in general it requires a large amount of computational resources.
In 1971 the present inventor co-authored a paper describing a computational method for determining the complete wave function (amplitudes and phases) from intensity recordings in the imaging and diffraction planes. See, “A Practical Algorithm for the Determination of Phase from Image and Diffraction Plane Pictures,” Cavendish Laboratory, Cambridge, England, Optik, Vol. 35, No. 2, (1972) pp. 237-246, which is incorporated herein by reference for background. The method depends on there being a Fourier Transform relation between the complex wave functions in these two planes. This method has proven to have useful applications in electron microscopy, ordinary light photography and crystallography where only an x-ray diffraction pattern may be measured.
The so-called Gerchberg-Saxton solution is depicted in a block diagram form in FIG. 1. The input data to the algorithm are the square roots of the physically sampled wave function intensities in the image 100 and diffraction 110 planes. Although instruments can only physically measure intensities, the amplitudes of the complex wave functions are directly proportional to the square roots of the measured intensities. A random number generator is used to generate an array of random numbers 120 between π and −π, which serve as the initial estimates of the phases corresponding to the sampled imaged amplitudes. If a better phase estimate is in hand a priori, that may be used instead. In step 130 of the algorithm, the estimated phases 120 (represented as unit amplitude “phasors”) are then multiplied by the corresponding sampled image amplitudes from the image plane, and the Discrete Fourier Transform of the synthesized complex discrete function is accomplished in step 140 by means of the Fast Fourier Transform (FFT) algorithm. The phases of the discrete complex function resulting from this transformation are retained as unit amplitude “phasors” (step 150), which are then multiplied by the true corresponding sampled diffraction plane amplitudes in step 160. This discrete complex function (an estimate of the complex diffraction plane wave) is then inverse Fast Fourier transformed in step 170. Again the phases of the discrete complex function generated are retained as unit amplitude “phasors” (step 180), which are then multiplied by the corresponding measured image amplitudes to form the new estimate of the complex wave function in the image plane 130. The sequence of steps 130-180 is then repeated until the computed amplitudes of the wave forms match the measured amplitudes sufficiently closely. This can be measured by using a fraction whose numerator is the sum over all sample points in either plane of the difference between the measured and computed amplitudes of the complex discrete wave function squared and whose denominator is the sum over all points in the plane of the measured amplitudes squared. When this fraction is less than 0.01 the function is usually well in hand. This fraction is often described as the sum of the squared error (SSE) divided by the measured energy of the wave function: SSE/Energy. The fraction is known as the Fractional Error.
A theoretical constraint on the above described Gerchberg-Saxton process is that the sum squared error (SSE), and hence the Fractional Error, must decrease or at worst remain constant with each iteration of the process.
Although the Gerchberg-Saxton solution has been widely used in many different contexts, a major problem has been that the algorithm can “lock” rather than decrease to a sum square error (SSE) of zero. That is to say, the error could remain constant and the wave function, which normally develops with each iteration, would cease to change. The fact that the SSE cannot increase may in this way trap the algorithm's progress in an “error well.” See Gerchberg, R. “The Lock Problem in the Gerchberg Saxton Algorithm for Phase Retrieval,” Optik, 74, 91 (1986), and Fienup, J. & Wackerman, C. “Phase retrieval stagnation problems and solutions,” J. Opt. Soc. Am. A, 3, 1897 (1986). All of the above-identified publications are hereby incorporated by reference for background. Another problem with the method became apparent in one dimensional pictures where non-unique solutions appeared. Furthermore, the algorithm suffers from slow convergence. To date, there are no alternative satisfactory solutions to these problems with the Gerchberg-Saxton method. Accordingly, there is a need for a system and method that can recover wave front phase information without the drawbacks associated with the prior art.